A new Kempe invariant and the (non)-ergodicity of the Wang-Swendsen-Kotecky algorithm

Abstract

We prove that for the class of three-colorable triangulations of a closed oriented surface, the degree of a four-coloring modulo 12 is an invariant under Kempe changes. We use this general result to prove that for all triangulations T(3L,3M) of the torus with 3<= L <= M, there are at least two Kempe equivalence classes. This result implies in particular that the Wang-Swendsen-Kotecky algorithm for the zero-temperature 4-state Potts antiferromagnet on these triangulations T(3L,3M) of the torus is not ergodic.

37 pages (LaTeX2e). Includes tex file and 3 additional style files. The tex file includes 14 figures using pstricks.sty. Minor changes. Version published in J. Phys. A

References in corpus (2)

• The multivariate Tutte polynomial (alias Potts model) for graphs and matroids
• Phase diagram of the chromatic polynomial on a torus

Cited by in corpus (6)

• Worm Monte Carlo study of the honeycomb-lattice loop model
• On the non-ergodicity of the Swendsen-Wang-Kotecky algorithm on the kagome lattice
• The three-state Potts antiferromagnet on plane quadrangulations
• The phase diagram for the bisected-hexagonal-lattice five-state Potts antiferromagnet
• Ergodicity of the Wang--Swendsen--Kotecký algorithm on several classes of lattices on the torus
• Invariants of winding-numbers and steric obstruction in dynamics of flux lines